# Pure Cubic Fields 1

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 Pure Cubic Fields 1 Previous Page § 4. Ambiguous principal ideals. Since the cyclotomic unit group Uk is generated by < -1, ζ, >, and since NormN|kUN ≥ NormN|kUk = Uk3 = < -1 >, we obtain bounds for the unit norm index (Uk:NormN|kUN) ∈ {1,3}, according to whether ζ ∈ NormN|kUN or not. The somewhat abstract quotient EN|k/UN1 - σ is isomorphic to the more ostensive quotient PNGal(N|k)/Pk of the group of ambiguous principal ideals of N|k modulo the subgroup of principal ideals of k, according to Iwasawa (or also to Hilbert's Theorems 92 and 94), and by the Takagi/Hasse Theorem on the Herbrand quotient, we have (PNGal(N|k):Pk) = (EN|k:UN1 - σ) = 3 (Uk:NormN|kUN) ∈ {3,9}. Even in the worst case (PNGal(N|k):Pk) = 3, there are at least the radicals D1/3, D2/3, and the unit 1 which generate 3 distinct ambiguous principal ideals of L|Q. Examples 4.1. We give the smallest radicands D of pure cubic fields L = Q(D1/3) where the two values of #Pr = (PNGal(N|k):Pk) actually occur: #Pr = 3 for D = 3 of type γ, #Pr = 9 for D = 2 of type β. § 5. Differential principal factorizations (DPF). As opposed to Hilbert's Theorem 94, the subgroup PN ∩ Ik/Pk ≤ PNGal(N|k)/Pk, the so-called capitulation kernel of N|k is trivial, because h(k) = 1. However, the elementary abelian 3-group Pr = PNGal(N|k)/Pk, whose generators are principal ideals dividing the different of N|k (so-called differential principal factors), consists of two nested subgroups, Pr ≥ Pa (similar to but not identical with our definitions in [Ma2]), absolute DPF of L|Q, Pa, always containing the above mentioned radicals, relative DPF of N|k, Pr \ Pa, such that the proper cosets of Pa in Pr do not project down to L|Q by taking the norm. The existence, resp. the lack, of certain prime divisors of the conductor f permits some criteria for the classification of pure cubic fields. Theorem 5.1. (Uk:NormN|kUN) = 1 can occur only when f is divisible by no other primes than 3 and / or primes qj ≡ ±1 (mod 9). Relative DPF of N|k can occur only if some prime qj ≡ +1 (mod 3) divides f. Absolute DPF of L|Q, distinct from radicals, must exist when no prime ≡ +1 (mod 3) divides f and f has at least one prime factor distinct from 3 and from primes ≡ -1 (mod 9). Remarks 5.1. 1. The condition for ζ ∈ NormN|kUN is due to the properties of the cubic Hilbert symbol (third power norm residue symbol) over k. 2. The condition for relative DPF of N|k is a consequence of the decomposition law (e,f,g) = (1,1,2) for primes ≡ +1 (mod 3) in k. Next Page